StudyBrixFree worksheets, study guides and courses

Physics · lesson 1 of 4

Rotational dynamics and rolling constraints

Angular velocity is the time derivative of angular displacement. For a fixed axis and constant moment of inertia, net torque equals I times angular acceleration.

University8 practice questionsNotes + worked examples
Take the quizStudy guideRevision sheet

Learn the ideas

Prerequisites: Newtonian mechanics, derivatives and energy.

Rotation

Angular velocity is the time derivative of angular displacement. For a fixed axis and constant moment of inertia, net torque equals I times angular acceleration.

Energy

Rigid-body rotation about a fixed axis has kinetic energy Iω²/2. A rolling body generally has both translational and rotational kinetic energy.

Constraints

For rolling without slipping, the centre speed equals Rω. That kinematic condition does not imply friction always dissipates energy.

Model limits

The moment of inertia depends on mass distribution and the chosen axis. A point-mass formula cannot replace the inertia of an extended object without justification.

Worked examples

Example 1

A fixed-axis rotor has I=3 kg m² and net torque 12 N m. Find its angular acceleration.

Use angular acceleration = torque/I = 12/3.

4 rad/s²

Example 2

A rotor with I=2 kg m² spins at 5 rad/s. Find its rotational kinetic energy.

K=Iω²/2=(2)(25)/2.

25 J

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 A fixed-axis rotor has I=3 kg m² and net torque 12 N m. Find its angular acceleration.rad/s²

Show answer and explanation

4 rad/s²

Use angular acceleration = torque/I = 12/3.

2 A rotor with I=2 kg m² spins at 5 rad/s. Find its rotational kinetic energy.J

Show answer and explanation

25 J

K=Iω²/2=(2)(25)/2.

3 A wheel of radius 0.4 m rolls without slipping at centre speed 2 m/s. Find ω.rad/s

Show answer and explanation

5 rad/s

The rolling constraint gives ω=v/R=2/0.4.

4 Which change definitely changes the moment of inertia about an axis?

Show answer and explanation

Moving some mass farther from that axis

Inertia is the mass-weighted squared distance from the specified axis.

5 Explain why a rolling solid cylinder has more kinetic energy than a point mass of equal mass moving at the same centre speed.

Sample answer and marking guidance

The cylinder has translational energy mv²/2 plus rotation about its centre Iω²/2. The point-mass model includes only translation.

Credit both terms and use of the same centre speed; do not claim that all rolling bodies have the same I.

6 Using I=mr²/2 for a solid cylinder, m=4 kg and centre speed 3 m/s, find its total rolling energy.J

Show answer and explanation

27 J

Translation is 18 J and rotation is mv²/4=9 J.

7 For a fixed axis, what sets the sign of angular acceleration when I is positive?

Show answer and explanation

The sign of net torque

With constant positive I, α=τnet/I.

8 State two assumptions behind the equation τnet=Iα in its simple scalar form.

Sample answer and marking guidance

Rotation is about the specified fixed axis and I is constant; use the torque component about that axis in an appropriate inertial treatment.

Award credit for the axis and constant-inertia assumptions rather than applying the relation to arbitrary deforming bodies.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.